Hi, kudos for making the effort to do all this!
According to Google, the dry weight of your Keystone 26BH is 6117 lbs. Not sure why your scaled wt. is less than this. The actual wt. of a trailer is typically around 1,000 lbs or more than the dry wt. As an example, our trailer is an ultralite, 29' long and has a factory dry wt. of 5237 lbs but has a loaded wt. of 6600 lbs. Some of our options account for part of this. If it hasn't been said previously, you should also have full propane tanks and the battery (or maybe 2 of them) in place along with all your normal cargo (BBQ, chairs, wheel chocks, food, clothing, etc. (and waste/water tanks must be empty). It shouldn't be that your scale weight is less than the dry weight unless your model is different or something is changed or different. Having full propane tanks or not won't make much of a difference on total trailer wt. but will affect the tongue weight.
But - assuming your trailer weight is in fact 5,960 lbs:
Your total trailer wt. is 11,720 - 5760 = 5,960 lbs.
First, your max. available payload is 1240 lbs (7,000 - 5760). Less the tongue wt. of 780 = 460 lbs. That's pretty good but does perhaps seem a bit off compared to other owner's vehicles. But assuming this is correct, you have 460 lbs left for DW, groceries, pets, camping gear and other stuff.
Your are correct, the numbers indicate that 200 lbs is transferred to your TV. This means that 580 lbs is being transferred to the trailer which is waaay too much. You should be looking at the weight on the TV front (steer) axle and rear (drive) axle individually. Weight onto the front axles is the key figure. Can't tell from your numbers so far though. I have read on another popular RV forum that a good distribution of tongue wt. is 25% onto steer axle, 50% onto drive axle and 25% onto trailer. So far, it looks like you have 75% going onto the trailer axles. This will in part explain why the handling does not feel good.
You want to try and get around 200 lbs just onto the front axles. Your bar rating of 1,000 lbs is okay. If round bars, you want to end up with 3 or 4 links left over on the chains. To get this may require some adjustment of the bar angle.
At this point, the wts show that you still need to transfer more weight onto the TV. This is also shown by the measured heights of your front fenders. I would suggest adjusting your bars so that the front fender height is at or near the original unloaded height, but not less. Then make another trip to the scales.
You are getting closer! ๐